Dissipative particle dynamics simulation of micro concentric eccentric annular flows

List of Figures Figure 5 The fully developed velocity profile compares with the Navior-Stokes solution in Poiseuille flow……….……...28 simple DPD fluid in Poiseuille flow……….……….…...29 Fig

DISSIPATIVE PARTICLE DYNAMICS SIMULATION OF MICRO-CONCENTRIC/ECCENTRIC ANNULAR FLOWS

PENGFEI CHEN

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2004

ACKNOWLEDGEMENTS

I am very grateful to my supervisors, Professor Nhan Phan-Thien and Associate

Professor Yeo Khoon Seng for giving me the great opportunity to study in such an

interesting area I would also like to express my sincere gratitude to them for their

constant guidance and encouragement throughout the course of this work

I would like to thank Professor Fan Xijun, Dr Dou Huashu, Dr Chen Shuo, Dr Lu

Zhumin, Dr Shi Xing, Mr Wu Tao, Ms Luo Chunshan and Ms Zhao Xijing for their

assistance and friendship

My entire family deserves a special gratitude for their unlimited support,

encouragement and love throughout my stay in NUS

Finally, I would like to give my acknowledgement to the National University of

Singapore for their Research Scholarship

Thanks are also due to all others who have helped me in one way or another in this

effort

Table of Contents

Acknowledgement I

Summary IV

Nomenclature V

List of Figures VIII

List of Tables XII

Chapter 1 Introduction 1

1.1 Background …… 1

1.2 Literature review …… 4

1.3 Objective of this research project …… 6

Chapter 2 Methodology 9

2.1 Basic equations of DPD method …… 9

2.2 Numerical Scheme …… 14

2.3 Initial Particle Models in concentric/eccentric flow field 16

Chapter 3 Steady Concentric/eccentric flows of simple DPD fluids at finite Reynolds numbers …… 20

3.1 Implementation of non-slip boundary condition in DPD 21

3.1.1 Boundary condition used in this thesis……… 24

3.1.2 Comparison of boundary conditions: Poisseuille flow… 26

3.1.3 Comparison of boundary conditions: Concentric rotating

3.2 Effects of some DPD Parameters on simulation……… 32

3.2.1 Effect of fluid particle density……….………33

3.2.2 Effect of conservative force factor between DPD particles……….……… 36

3.3 Steady Circular Couette flow and eccentric flow of simple DPD fluids at finite Reynolds numbers……… ……… 43

3.3.1 Circular Couette flow of simple DPD fluids at finite Reynolds numbers……….… 43

3.3.2 Eccentric flow (ωout/ωin=0) of simple DPD fluids at finite Reynolds numbers……….… 48

Chapter 4 Steady Circular/eccentric flows of FENE Chain Suspension at finite Reynolds numbers……….…………60

4.1 Circular Couette flow of FENE chain suspension at finite Reynolds numbers……… 62

4.2 Eccentric flow (ωout/ωin=0) of FENE chain suspension at finite Reynolds numbers……… ……… 70

Chapter 5 Conclusion and future works …… ….80

5.1 Conclusion………… ……… …80

5.2 Future works……….……….81

Appendix A……….……… … 84

Appendix B……… … 87

Reference………88-92

Summary

Dissipative Particle Dynamics (DPD) is a fairly new method for simulating complex

fluid flows and other colloidal phenomena It is a mesoscopic method and offers the

possibility of capturing some degree of molecular-level detail while conforming to

continuum hydrodynamics at larger length scales In this thesis, a new implementation

of the no slip boundary condition in the modeling of solid boundaries is studied This

boundary is implemented to simulate the planar Poisseuille and circular Couette flow,

and the results compare excellently with similar results derived by more traditional

CFD methods The effects of two important DPD parameters (particle density and

conservative coefficient) are studied It is shown that these two parameters affect the

simulation accuracy considerably and should be carefully set Furthermore, to confirm

the ability of DPD method to provide numerically accurate results in simulating

complex flow with rather complicated boundary conditions, the DPD method is

employed to study the flow behavior of three dimensional microscopic

concentric/eccentric flows at finite Reynolds numbers A simple DPD fluid (made up

of simple DPD particles) and then bio-molecular suspensions (FENE chains are used to

model DNA macromolecules) are studied in detail respectively

Nomenclature

It is not practical to list all the symbols that have been used Below the author list the

more important ones Some of the symbols are defined as they are used There are also

occasions where the same symbol is assigned a different meaning in a different context,

but the meaning should normally be clear from the usage

Rin Radius of inner cylinder

ρ Fluid density

c Mean annular gap width, Rout-Rin

List of Figures

Figure 5 The fully developed velocity profile compares with the Navior-Stokes

solution in Poiseuille flow……….…… 28

simple DPD fluid in Poiseuille flow……….……….… 29

Figure 10 The comparison of pressure profiles in Poiseuille flow……… 30

Figure 11 The comparison of velocity profiles of a simple DPD fluid in circular

Couette flow……….…32

Figure 12 Illustration of the effect of fluid particle density on velocity of a simple

DPD fluid in circular Couette flow……….….34

Figure 13 Comparison of temperature profiles of a simple DPD fluid in circular

Figure 14 Comparison of density profiles of a simple DPD fluid in circular Couette

Figure 15 Comparison of velocity profiles of a simple DPD fluid in circular Couette

circular Couette flow……….……….… 41

Figure 17 Comparison of temperature profiles of a simple DPD fluid in circular

Figure 18 Comparison of velocity profiles of a simple DPD fluid in circular Couette

Figure 19 Geometry for Circular Couette flow……….… 43

Figure 20 Simulated Vx, Vz velocity and streamline contours of a simple DPD fluid

in circular Couette flow……… 45

Figure 21 The fully developed tangential velocity profile of a simple DPD fluid in

circular Couette flow……….……… 46

Figure 22 Density and temperature profiles of a simple DPD fluid in circular Couette

flow……… …46

Figure 23 Pressure profile of a simple DPD fluid in circular Couette flow…….…47

Figure 24 Shear stress distribution of a simple DPD fluid in circular Couette

flow……… 47

Figure 25 Profiles of the first and second normal stress differences of a simple DPD

fluid in circular Couette flow……… ….48

Figure 26 Bipolar coordinate system and geometric parameters of the eccentric

annular region……… 50

Figure 27 Streamline patterns of a simple DPD fluid in eccentric flow (ωout/ωin=0)

with different Reynolds numbers……….………….58

Figure 28 Comparison of polar angles of separation and reattachment points of

Figure 31 Comparison of velocity profiles of FENE chain suspensions of different

concentrations in the Circular Couette flow……….…….64

Figure 32 Comparison of pressure profiles of FENE chain suspensions of different

concentrations in the Circular Couette flow……….……….65

Figure 33 Comparison of Shear stress profiles of FENE chain suspensions of

different concentrations in the Circular Couette flow……….…… 65

Figure 34 The conformation of some typical FENE chains in circular Couette flow

(ωout/ωin=0) with different volume fractions……….………….73

Figure 39 Illustration plot of the development of the back flow of FENE chain

Figure 40 Snapshots showing the detailed backflow of one FENE chain in eccentric

flow……… 79

List of Tables

fractions……… 73

Chapter 1 Introduction 1.1 Background

The numerical simulation of hydrodynamic interactions between suspended particles

and the surrounding fluid phase is of interest in many engineering applications

associated with particle transport, such as colloids, polymers, aerosols and

physiological system [1] The properties of these systems are often determined by

their mesoscale structures, i.e between the atomistic scale and the macroscopic scale,

thus endowing a complex fluid with unique and interesting features [2]

Dynamic simulation of these systems presents unique problems that are difficult to

address with established methods On the one hand, the time and length scales of

interest make simulation by molecular dynamics (MD) impractical, a limitation that is

not likely to be solved in the near future, even with the rapid increases in

computational speed that are expected to develop On the other hand, colloidal length

scales are often almost the same order as the flow domain size (The ratio of the

colloidal length to the characteristic length of flow field can be taken as the equivalent

continuum) So purely continuum approaches, such as conventional computational

fluid dynamics (CFD), are unacceptable, requiring as they do that one leaves out so

many details at the molecular level Moreover, people still cannot make the full

connection from atomistic length-scale to the macroscopic world Hence to obtain a

better understanding of the phenomena that occur in mesocale, some intermediate

simulation techniques are developed that are aimed at a length-scale larger than the

atomistic scale, but smaller than the macroscopic scale Dissipative particle dynamics

(DPD) is a so-called mesoscopic simulation method that provides one possible means

of bridging the gap between purely molecular and continuum-level treatments

Dissipative particle dynamics is a stochastic simulation technique introduced by

Hoogerbrugge and Koelman [3] in 1992 to simulate complex fluid dynamical

phenomena DPD combines features from molecular dynamics and lattice-gas

automata (LGA) by introducing a LGA-type of time-stepping into MD schemes In

contrast to molecular dynamics simulations, the particles are supposed to represent the

fluid on a mesoscopic level rather than a molecular level For the simulation of

macroscopic fluid dynamic phenomena, this implies an advantage of computational

effort [4] Though Brownian Dynamics Simulation (BDS), LGA and Lattice

Boltzmann (LB) are mesoscale simulation methods, it is difficult for BDS to deal with

complex flow field and for LGA and LB to cope with complex fluids and DPD has the

advantage of more flexibility

The basic unit in DPD system is a set of discrete momentum carriers called particles

that move in continuous space and discrete time-steps The momentum carriers are

coarse grained entities which are no longer regarded as molecules in a fluid but rather

representing the collective dynamic behavior of a large number of molecules (a fluid

Particle Dynamics is such a method for simulation of the motion of this kind of “fluid

particles” often referred to in continuum mechanics Conceptually this method

amounts to dividing up the molecules of a flow field into groups or packets, which

have dimensions many times larger than the mean free path of an individual molecule

The mass m of each packet is localized to a point As the points move, their interaction

is ‘soft’ as a result of the fact that the packets are deformable, and this deformation is

accompanied by a dissipation of energy In addition, because of their small

colloid-like dimensions, the thermal motion of the molecules in the packets gives rise

to a random or Brownian contribution to their motion A time step in a DPD

simulation therefore consists of summing interactions that consist of three terms: a

conservative repulsion that accounts for steric and energetic interactions between the

molecules of interacting packets and this repulsion force prevents particles from

overlapping [5], a dissipative interaction that accounts for energy that is lost due to

internal friction or viscosity within a packet, and a random step that arises from the

collective thermal motion of the molecules within a packet The interactions are

summed over all pairs of particles, in a way that guarantees that linear and angular

momentum are conserved Unless an explicit connection to molecular-level

interactions is needed, the parameters that govern these interactions are chosen in any

way that reproduces the continuum-level dimensionless groups that determine the

behavior of a particular system

The last two terms, which account for dissipation and random motion, are necessarily

coupled by a fluctuation-dissipation theorem and the principle of equipartition of

energy These two terms combine to create a continuous pseudofluid in which the

particles are suspended and free to interact hydrodynamically The original algorithm

of Hoogerbrugge and Koelman [3] did not satisfy this requirement, leaving in doubt

whether a simulation could reach a true equilibrium, even at long times and in the

absence of bulk motion A slight revision of the original algorithm, produced by

Espanol and Warren [6], remedied this problem, and has been used in most

applications since that time The method has received considerable theoretical support

in other areas as well Marsh et al [7] make explicit connections between DPD and

the Navier-Stokes equations by deriving the Fokker-Planck-Boltzmann equation for

the single-particle distribution function, and solving it by the Chapman-Enskog

method Their derivation includes expressions for transport properties such as the

shear viscosity that are valid in the limit of strong damping, where conservative

repulsive interactions are negligible Flekkoy and Coveney [8] discussed creating

DPD particles by grouping molecules in MD simulations together

1.2 literature review

Since DPD method is a fairly recent development and the method is still evolving,

only the fundamental and representative papers will be reviewed here Literature

review about the flow field in concentric/eccentric flows will be noted in the section

on results and discussion

A key and attractive feature of DPD is its ability to reproduce continuum-level fluid

mechanics over large enough length scales, even in the presence of inertia However,

in spite of the growing literature on the applications of DPD to various problems,

there are still relatively few direct, quantitative comparisons between calculations

done with DPD and well-established analytical and numerical results Those

comparisons that do exist pertain exclusively to low Reynolds numbers

Simulation results obtained by DPD have also been reported for several problems of

interests In their original paper, Hoogerbrugge and Koelman [3] calculated the

flow-induced drag on a cylinder in a periodic array, and compared it with result

reported by Sangani and Acrivos [9] This comparison was made in the limit of low

Reynolds number, where inertial effects in the flow are negligible Boek et al [10, 11]

studied the rheological properties of colloidal suspensions of spheres and rods using

dissipative particle dynamics, and measured the viscosity as a function of shear rate

and volume fraction of the suspended particles Furthermore Boek and van der Schoot

[12] used DPD to study fluid flow through a periodic array of cylinders as a model for

fluid filtration through a porous medium, and discussed the resolution effects in DPD

Both the calculations of Hoogerbrugge and Koelman [3] and those of Boek et al [10,

11, 12] were done with the original algorithm, without the modification proposed by

Espanol and Warren [6]; Boek et al [12] argued that the modification does not

significantly alter the calculations of suspension rheology Model polymers have been

constructed by linking DPD particles together with springs, and polymer dynamics

and associated parameters have been studied by Kong et al [13] In addition,

polymer-surfactant aggregation and micelle formation have been simulated by Groot

[14], and an extension of DPD to non-Newtonian flows (Bosch [4]) has been proposed

More recently, Fan et al [15] presented their simulation results for macromolecular

suspension flows through microchannels They also studied the Poiseuille flow of

simple DPD fluids and FENE (Finitely Extendable Nonlinear Elastic) chains

suspension and found that simple DPD fluids behave just like a Newtonian fluid while

FENE chains suspension can be fitted by dilute suspensions

1.3 Objective of this research project

As noted above, DPD has its own advantages to numerically simulate the

hydrodynamic interactions of various complex systems: such as polymer suspensions

[13, 16, 17, 18], colloids [10, 11, 19] and multiphase fluids [20, 21, 22] Although it is

a promising technique for complex fluids, there are very few microfluidic applications

of DPD in complex systems reported The most recent literature that combines such a

microscopic application of DPD in complex system as well as giving a quantitative

simulation is noted in the paper of Fan et al [15], but what is studied in their paper is

about the simple Poiseuille flow In this thesis, a more complex flow problem is

studied: steady microscopic concentric/eccentric flow at finite Reynolds numbers

This work is also stimulated by the recent innovations in MEMS devices, especially in

understood today

Unlike the Poiseuille flow, firstly, the boundary condition of concentric/eccentric flow

is more complicated because it is a closed flow field and the implementation of the

boundary conditions will have a more significant impact on the simulation results

Secondly, the flow patterns in concentric/eccentric flows are more intricate compared

with that of Poiseuille flow at low Reynolds numbers, especially for eccentric flow

(there is a clear recirculation region present in the flow field) Same with what are

presented in the paper of Fan et al., two kinds of fluids are studied here: simple DPD

fluids and FENE chains suspension Some results on the deformation and migration of

FENE are also reported

It should be noted that the purpose of this research program is not to propose DPD as

an instrument for use in computational fluid mechanics, but rather to test its ability to

capture continuum fluid mechanical effects in the presence of significant inertia Even

capture inertial effects accurately is important In their study of the Brownian motion

corrections to the equations of motion for each frequency of oscillation They found a

nonlinear force of interaction between the particles that, even for very small Reynolds

and temperature, and R is the particle radius Such an interaction has an effect on the

distribution of the particles, which in turn affects the thermodynamics and rheological

properties of a colloidal suspension

Since fluid simulation by DPD is a fairly new development, the results presented in

this thesis are focused on a number of test problems These problems include:

improving the dynamic behavior of DPD method, studying the effects of basic DPD

parameters (α and n) on simulation results, and accessing the ability of DPD to

provide numerically accurate results in simulating complicated flows with

complicated boundary conditions The simulation results presented in this work

confirm the ability of DPD to provide quantitatively accurate results for complicated

flows, provided conditions are such that the compression of the DPD fluid is not

significant

In the present work, the DPD method is used to calculate three-dimensional flows at

finite Reynolds numbers, and examine conditions under which Newtonian fluid and

non-Newtonian fluid behaviour is reproduced quantitatively First, the DPD method is

outlined This is followed by the description of a new implementation of DPD

boundary condition, and then the effects of DPD parameters are studied, with some

simulation details presented The simulation details for concentric/eccentric flows

with simple DPD fluids and FENE chains suspensions are presented next

Chapter 2 Methodology 2.1 Basic equations of DPD method

The DPD system consists of a set of interacting “particles”, whose time evolution is

mass is taken to be the mass of a particle, so that the force acting on a particle is

particle i by particle j , which is assumed to be pairwise additive The dynamic

interactions between the particles are composed of two parts, dissipative and

stochastic, complementing each other to ensure a constant value for the mean kinetic

ij ij j

i ij

ij F F F

≠

(2)

Since the time average of the dissipative and fluctuation forces is zero, they do not

feature in the equilibrium behavior of the system, which is governed solely by

along the line of centers and is given by

C

ˆ(1 / ) ( )

ij

ˆ( )(ˆ )

ij = −γw r ij ij⋅ ij ij

ij= −i j

characterizing the extent of dissipation in a single simulation step The negative sign

step:

( ) 0

ij t

over a time scale considerably larger than its correlation time scale:

The detailed balance condition, similar to a Fluctation-Dissipation theorem relating

the strength of the random force to the mobility of a Brownian particle, requires that

will ensure that the temperature remains constant

We use the following weight function to improve on the Schmidt number for the

This weight function yields a stronger dissipative force between particles than that

from the standard quadratic force for a given configuration of particles and

interaction strength

ij

F ,

hand, reduces the relative velocity of two particles and removes kinetic energy from

their mass centre to cool the system down When the detailed balance is reached, the

system temperature will approach the given value The dissipative and random

forces act like the thermostat in molecular dynamics (MD)

When simulating complex fluids and flows, simple DPD particles, described above,

are used to model solvent or suspending fluid The solid walls can be modeled by

frozen DPD particles The Finitely Extendable Nonlinear Elastic (FENE) chain is a

model commonly used to model flexible polymer molecules in rheology; it is used in

this thesis to model bio-molecules Beads of the polymer chain are replaced by DPD

particles The intermolecular forces will act on these particles and should be added to

the right hand side of Eq (1)

In the FENE chain [24], the force on bead i due to bead is j is

2

ij S

/

ij m

predict a shear-rate dependent viscosity and finite elongational viscosity The mass

of the beads is assumed to be unity, which is as same as that of other simple DPD

fluid particles

The time constant is important in characterizing molecular motion and can be

formed from model parameter Two constants with the time dimension can be

obtained for the FENE spring [24],

k T

λλ

Chain models usually have a spectrum of relaxation times [24] There is no closed

form expression for the relaxation time spectrum for FENE chains; however, a

modification of FENE chain, called the FENE-PM chain, has the same spectrum as

the Rouse chain (bead and Hookean spring chain) [24] A time constant can be

defined for FENE-PM chain as [25]:

2

1

b fene H

N b

b

The contour length is an appropriate parameter to represent the molecular size If

c

m c

r L

b

=

2.2 Numerical Scheme

The initial configuration of fluid and wall particles are generated separately by a

pre-processing program and read in as input data The total number of particles

depends on the size and geometry of the flow domain, the densities of the fluid and

wall materials The initial velocities of fluid particles are set randomly according to

the given temperature but the wall particles are frozen At the beginning of the

simulation the particles are allowed to move without applying the external force and

rotation until the thermodynamic equilibrium state is reached Then the rotating

velocity is applied to the inner or outer cylinder wall particles and the

non-equilibrium simulation starts

The forces computation and time updates are conducted with Cartesian coordinate

system and the samples averaging work are carried out with polar coordinate system

for concentric flow and double-polar coordinate system for eccentric flow

Since the dissipative force is dependent on the velocity, a modified version of the

velocity-Verlet algorithm [26] is used This algorithm can be described as follows:

which account for some additional effects of the stochastic interactions If the total

force is velocity independent, the standard velocity-Verlet algorithm is recovered for

λ =0.5 Groot and Warren [26] found that the optimum value of λ is 0.65 For this

equation describes the contribution to the stress from the momentum transfer of DPD

particles and the second term from the interparticle forces For simple DPD particle

13

p= − trS (21)

2.3 Initial Particle Models in concentric/eccentric flow field

Since we assume that the purpose of the simulation is to study the equilibrium fluid

state, then the nature of the initial particle configuration should have no influence

whatsoever on the outcome of the simulation In choosing the initial coordinates, the

usual method is to position the atoms at the sites of a lattice whose unit cell size is

chosen to ensure uniform coverage of the simulation region Typical lattices used in

three dimensions are the face-centered cubic(FCC) and simple cubic, whereas in two

dimensions the square and triangular lattices are used; if the goal is the study of the

solid state, then this will dictate the lattice selection There is little point in

laboriously constructing a random arrangement of atoms, typically using a Monte

Carlo procedure to avoid overlap, since the dynamics will produce the necessary

randomization very quickly An obvious way of reducing equilibration time is to

base the initial state on the final state of a previous run

The function of FCC arrangement (with the optional of unequal edges) is to generate

four particles per unit cell To generate more particles in one cell, there are other

arrangement methods For example, the diamond lattice, which can generate eight

particles per unit cell, is a slightly more complicated form of the FCC code since the

lattice is most readily defined as two staggered FCC lattices, one of which is offset

along the diagonal by a quarter unit cell Similarly, to distribute 12 particles in one

unit cell, three staggered FCC lattices can be built up with offset of 0.125 unit cell

These three arrangements are used in this thesis to generate the required particle

(1) Fluid Particle disposition

First, particles are distributed evenly in a prism whose square section is bigger than

the outer circle; then deduct particles outside of the outer circle and particles within

the inner circle, and the shade area shown in Figure 1 represents the needed flow

domain

In this three-dimensional problem, the periodic boundary condition is applied to

fluid boundaries of the axis direction, which is y direction of Figure 1 Particles that

leave the simulation region of y direction immediately reenter the region through the

opposite face For this steady, low Reynolds number case, this application of

periodic boundary condition is workable and feasible

Figure 1 Illustration of generating fluid particles in an annular area

(2) Wall Particle disposition

Solid walls are modeled by frozen DPD particles; three wall layers with staggered

distribution in angular (Fig.2) and radial (Fig 3) directions

Due to the soft interaction between DPD particles, it is difficult to prevent particles

from penetrating the wall if the wall particle density is too low compared with the

fluid particle density; however, a higher density of the wall will produce a stronger

repulsive force between the wall DPD particles and fluid DPD particles, and

consequently a large density fluctuation appears near the wall So normally the wall

particle density is chosen to be of the same order of that of fluid particles

X

Y

Z

On the other hand, the repulsive force between wall particles and fluid particles is

also determined by the value of repulsive force coefficient between them So the

wall particle density should be set rationally, although there is no certain mode to

distribute the wall particles for such a special wall boundary contour

Figure 2 Illustration of generating wall

particles in angular direction

Figure 3 Illustration of generating wall particles in radial direction

Chapter 3 Steady Concentric/eccentric flows of simple

DPD fluids at finite Reynolds numbers

Some of implementation details on DPD are briefly introduced here The program is

usually divided into 2 parts: a pre-processing program and a main one In

pre-processing program, the initial particle configurations for the fluid, wall and

DNA chains (if any) are generated and read into the main program as input data

When the density of fluid and the geometry of flow domain are defined, the wall and

DNA chains particles if any will be set, and then all fluid particles are initially

located at the sites of a face-centered cubic (FCC) lattice

In the main program, the initial velocities of fluid particles are set randomly

according to the given temperature The velocities of wall particles are set at zero At

the beginning of simulation the fluid particles are allowed to move without applying

any external forces until a thermodynamic equilibrium state is reached Then the

external force field is applied to fluid particles and the non-equilibrium simulation

starts

In this kind of simulation, computing the interaction forces between particles takes

the most computational time To cope with this problem, here a cell sub-division and

linked-list method described by Rapaport [28] is applied In this method, firstly the

particles with the cells in which they reside at any given instance The linked list

therefore associates a pointer with each data item and to provide a non-sequential

path through the data Thus it needs only a one dimension array to store the list and

makes the search of neighboring particles more efficiently

Three parts will be discussed below: firstly the boundary conditions used in this thesis

are outlined, afterwards the effects of two DPD parameters on the simulation are

analyzed, then the simulation results of simple DPD particles in concentric/eccentric

flows are presented The simulation results of DNA polymer chains suspension in

concentric/eccentric flows will be presented in the next chapter

3.1 Implementation of non-slip boundary condition in DPD

As with any other methods in computational fluid dynamics, the issue of boundary

conditions has to be addressed in DPD Until now, the boundary conditions in DPD

are usually treated in two ways:

1 In order to simulate a shear flow such as that generated in a Couette geometry,

the Lees-Edward technique has been used [3, 6, 19], which is a way of avoiding

the modelization of physical boundaries

2 When considering the boundary conditions on the surface of solid objects as in

the modelization of colloidal suspensions, the method of “freezing” some

portions of the fluid described by DPD particles has been used [7, 29, 30, 31] In

this model, those particles are fixed but still can interact with other free particles

[32]

Due to the soft repulsion between DPD particles, it is difficult to prevent fluid

particles from penetrating the wall particles Near-wall particles may not be slowed

down enough and slip may then occur To avoid this, higher density of wall particles

and larger repulsive forces have to be adopted to strengthen the wall effects This,

however, results in large density distortions in the flow field, which is similar to what

happened in MD simulation Special treatments have been proposed to implement no

slip boundary condition in DPD simulation without using frozen wall particles

Revenga et al [33, 34] used effective forces to represent the effect of wall on fluid

particles instead of using wall particles For planar wall, the effective forces can be

obtained analytically But these forces are not sufficient to prevent fluid particles from

crossing the wall When particles cross the wall, a wall reflection is used to reflect

particles back to the fluid However, a slip velocity appears at the wall if the wall

density is equal to that of the fluid, and no slip results only if the wall has a much

higher density than that of the fluid To cope with this problem, Willemsen et al [35]

added an extra layer of particles outside of the simulation domain The position and

velocity of particles in this layer are determined by the particles inside the simulation

domain near the wall, such that the mean velocity of a pair of particles inside and

outside the wall satisfies the given boundary conditions Furthermore, to reduce the

the particles being located between r c and 2r c from the boundary into this layer, and

considering the repulsive forces between those particles and free particles Obviously,

this method is quite expensive computationally and very difficult to use for the

complex surface of solid objects such as a sphere or cylinder

Since the effective forces are not sufficient for keeping the fluid confined and one has

to specify what happens when a DPD particle crosses the line that defines the position

of the wall Normally, there are three different possibilities:

1 Specular reflection: such that the parallel component of the momentum of the

particles is conserved and the normal components is reversed

2 Maxwellian reflection: where the particles are introduced back into the system

according to a Maxwellian distribution of velocities centered at the velocity of

the wall

3 Bounce back reflection: in which both components of the velocity are reversed

In [34], Revenga et al compared these three methods, and they found that for small

is the thermal velocity), slip appears in specular and Maxwellian reflections and the

the bounce back scheme produces an anomalous temperature behavior

Later, considering the merits and disadvantages of above methods, Fan et al [15]

present their model based on the above discussions and obtained good simulation

results for Posieuille flow They still used frozen particles to represent the wall, but

near the wall a thin layer is assumed where the no slip boundary condition holds To

meet this boundary condition, they enforced a random velocity distribution in this

layer with zero mean and corresponding to the given temperature Similar to Revenga

et al.’s reflection law, Fan et al [15] further required that particles within this layer

always leave the wall The velocity of particle i in the layer is

into the fluid domain The above equation means that the normal velocity component

of fluid particles within the assumed thin layer will always move away from the wall

and the parallel component of the momentum of the particle is conserved

In this section, a new boundary condition model of the frozen particles to represent

the wall is presented which satisfies no-slip boundary condition and also has its own

merits Firstly, this boundary condition model is outlined, and then simulation results

from this boundary condition are compared with those of Fan et al.’s model [15] in

two cases: Poisseuille flow and circular Couette flow

3.1.1 Boundary condition used in this thesis

interior; In order to produce a non slip effect, each particle colliding with the wall has

all memory of its previous velocity erased, and is reflected back into the system with a

new velocity having a random direction and fixed magnitude which is set to a value

corresponding to the wall temperature In addition, for the wall of the rotating

cylinder the local (tangential) wall velocity is added to yield a new velocity vector

This mechanism is sufficient to drive the fluid rotation and dissipate the thermal

energy generated by the sheared flow

The main differences from Fan et al’s model [15] lie in:

1 Only particles which intend to cross the wall boundary instead of all particles

lying within the thin layer are imposed with a new velocity corresponding to

the given temperature;

2 In addition, the updated positions of those reflected particles are traced

accurately in the program rather than being given an approximate position as

noted by Rapaport [28] This idea can be shown clearly with Figure 4:

Different with Maxwellian reflection scheme (noted above), in which the new

velocity is distributed with the random directions, the boundary condition used here

retains the parallel part of the old velocity and makes the vertical vector point into the

interior area (see Equation (22))

(a) t0→ + ∆t0 λ t (0< < with the λ 1)

original velocity

the new velocity

Figure 4 Illustration of the boundary reflection

3.1.2 Comparison of boundary conditions: Poisseuille flow

Fan et al.’s boundary model [15] gives a rather good simulation results for simple

Poisseuille flows, but in their article an obvious fluctuation of density and pressure in

the region near the wall was noted, although this fluctuation is not as severe as what

was predicted by Molecular Dynamics simulation In this section, to make a

convincing comparison, the same parameters are selected as noted in [15] The

wall particles are located in three layers parallel to the (x, y) plane in each side The

boundary condition is applied to fluid boundaries in the X and Y directions The

gravity force g=0.02 is applied to drive the flow The simulation region is divided into

300 bins in the z- direction, of which 278 bins are located in the fluid domain, and

others are in the wall domain All local flow properties are obtained by averaging the

In Figure 5, 6, 7 and 8, the fully developed velocity, temperature, shear stress and

normal stress (N1 and N2) profiles are depicted Similar to what have been presented

in [15], these profiles simulated with the new boundary condition coincide excellently

with the analytical results We could find that the velocity and shear stress profiles

agree with the Navier-Stokes analytical solution accurately except for the region near

the wall surfaces; the temperature profile is uniform across the channel; the first (N1)

and second (N2) normal stress difference profiles are almost zero along the z-

direction which confirms that the simple DPD particle solvent is a Newtonian fluid,

which is also a main conclusion of Fan et al.’s paper [15]

While in Figure 9, we could find the density fluctuation near the wall region is

reduced significantly (about 40%) with the new boundary condition As a result, the

distortion of the pressure is also alleviated greatly as shown in Figure 10 So with the

new implementation for modeling the boundary, a better simulation result is obtained

without increasing the computation effort If the wall particle density is retained, a

better result may be reached if we find out more precise parameter such as the